Quasi-Hopf twistors for elliptic quantum groups

Abstract

The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [FIJKMY1], Felder [Fe]). Fronsdal [Fr1, Fr2] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebraU q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universalR matrix ofU q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. This construction entails that, for generic values of the deformation parameters, the representation theory forU q(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebraA q,p ( $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$ ).